Using differentials, find the approximate values of the following:

(0.009)1/3

Let us assume that


Also, let x = 0.008 so that x + Δx = 0.009


0.008 + Δx = 0.009


Δx = 0.001


On differentiating f(x) with respect to x, we get



We know





When x = 0.008, we have







Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as



Here, and Δx = 0.001


Δf = (8.3333)(0.001)


Δf = 0.0083333


Now, we have f(0.009) = f(0.008) + Δf




f(0.009) = 0.2 + 0.0083333


f(0.009) = 0.2083333


Thus, (0.009)1/3 ≈ 0.2083333


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